Supercongruences for polynomial analogs of the Ap\'ery numbers

We consider a family of polynomial analogs of the Ap\'ery numbers, which includes $q$-analogs of Krattenthaler--Rivoal--Zudilin and Zheng, and show that the supercongruences that Gessel and Mimura established for the Ap\'ery numbers generalize to these polynomials. Our proof relies on polynomial analogs of classical binomial congruences of Wolstenholme and Ljunggren. We further indicate that this approach generalizes to other supercongruence results.